Solving Large-Scale Nonlinear System of Equations

Description

Usage

Arguments

a function that takes a real vector as argument and returns a real vector of same length (see details).

par

A real vector argument to fn, indicating the
initial guess for the root of the nonlinear system.

method

An integer (1, 2, or 3) specifying which Barzilai-Borwein steplength to use. The default is 2. See *Details*.

control

A list of control parameters. See *Details*.

quiet

A logical variable (TRUE/FALSE). If TRUE warnings
and some additional information printing are suppressed.
Default is quiet = FALSE
Note that the control variable trace and quiet
affect different printing, so if trace is not set to FALSE
there will be considerable printed output.

alertConvergence

A logical variable. With the default TRUE
a warning is issued if convergence is not obtained. When set to FALSE
the warning is suppressed.

...

Additional arguments passed to fn.

Details

The function dfsane is another algorithm for implementing non-monotone
spectral residual method for finding a root of nonlinear systems, by working
without gradient information.
It stands for "derivative-free spectral approach for nonlinear equations".
It differs from the function sane in that sane requires an
approximation of a directional derivative at every iteration of the merit
function F(x)^t F(x).

R adaptation, with significant modifications, by Ravi Varadhan, Johns Hopkins University (March 25, 2008), from the original FORTRAN code of La Cruz, Martinez, and Raydan (2006).

A major modification in our R adaptation of the original FORTRAN code is the availability of 3 different options for Barzilai-Borwein (BB) steplengths: method = 1 is the BB
steplength used in LaCruz, Martinez and Raydan (2006); method = 2 is equivalent to the other steplength proposed in Barzilai and Borwein's (1988) original paper.
Finally, method = 3, is a new steplength, which is equivalent to that first proposed in Varadhan and Roland (2008) for accelerating the EM algorithm.
In fact, Varadhan and Roland (2008) considered 3 similar steplength schemes in their EM acceleration work. Here, we have chosen method = 2
as the "default" method, since it generally performe better than the other schemes in our numerical experiments.

Argument control is a list specifing any changes to default values of algorithm control parameters. Note that the names of these must be
specified completely. Partial matching does not work.

M

A positive integer, typically between 5-20, that controls the monotonicity of the algorithm. M=1 would enforce strict monotonicity
in the reduction of L2-norm of fn, whereas larger values allow for more non-monotonicity. Global convergence under non-monotonicity is ensured by
enforcing the Grippo-Lampariello-Lucidi condition (Grippo et al. 1986) in a non-monotone line-search algorithm. Values of M
between 5 to 20 are generally good, although some problems may require a much
larger M. The default is M = 10.

A logical variable (TRUE/FALSE). If TRUE, information on
the progress of solving the system is produced.
Default is trace = TRUE.

triter

An integer that controls the frequency of tracing when
trace=TRUE. Default is triter=10, which means that
the L2-norm of fn is printed at every 10-th iteration.

noimp

An integer. Algorithm is terminated when no progress has been
made in reducing the merit function for noimp consecutive iterations.
Default is noimp=100.

NM

A logical variable that dictates whether the Nelder-Mead algorithm
in optim will be called upon to improve user-specified starting value.
Default is NM=FALSE.

BFGS

A logical variable that dictates whether the low-memory
L-BFGS-B algorithm in optim will be called after certain types of
unsuccessful termination of dfsane. Default is BFGS=FALSE.

Value

A list with the following components:

par

The best set of parameters that solves the nonlinear system.

residual

L2-norm of the function at convergence,
divided by sqrt(npar), where "npar" is the number of parameters.

fn.reduction

Reduction in the L2-norm of the function from the
initial L2-norm.

feval

Number of times fn was evaluated.

iter

Number of iterations taken by the algorithm.

convergence

An integer code indicating type of convergence. 0
indicates successful convergence, in which case the resid is smaller
than tol. Error codes are
1 indicates that the iteration limit maxit has been reached.
2 is failure due to stagnation;
3 indicates error in function evaluation;
4 is failure due to exceeding 100 steplength reductions in
line-search; and
5 indicates lack of improvement in objective function
over noimp consecutive iterations.

R Varadhan and PD Gilbert (2009), BB: An R Package for Solving a Large System of Nonlinear Equations and for Optimizing a High-Dimensional Nonlinear Objective Function, J. Statistical Software, 32:4, http://www.jstatsoft.org/v32/i04/